1. Field of the Invention
The present invention relates to methods and systems for operational scheduling of facilities, such as electrical energy generation facilities, and valuation of facilities and assets such as generation assets, and more particularly, to methods and systems for determining optimal operational scheduling for facilities, for determining anticipated profitability of facilities, and for determining values of assets such as generation assets.
2. Description of the Related Art
Electrical power generation facilities generally operate to convert fuel into electrical energy for profit. As such, the facilities are a type of energy generation asset, and their value at a given time is based on their anticipated profitability. Investing and trading in generation assets is possible through certain forms of financial instruments, such as real option contracts, whose value can be linked to a generation facility (or facilities), its operation, or the power produced by the facility. Particularly, the value of such instruments is often associated with the profitability of operation of a facility over a specified period of time. Such profitability can be affected by many factors, including price paths for fuel and other relevant commodities, as well as scheduling of operation of the facility, including, for example, generation rate increases and decreases made during the period. A profitability assessment should naturally assume that an attempt will be made, throughout the period of time, to schedule operation of the facility so as to optimize its profitability. Of course, determining optimal scheduling of a facility is important in its own right for scheduling decision-making, as well as being necessary for accurate valuation of generation assets and associated financial instruments. The important tasks of determining optimal scheduling as well as valuing generation facilities pose difficult challenges, due in part to the large number of interdependent variables involved, as well as the uncertainties of such things as relevant future price paths.
Optimal scheduling of a generation facility over a period of time is influenced by many variable factors. One set of factors that can influence optimal scheduling includes prices at various times during the period, or price paths, of commodities used by the facility at a rate dependent on the rate of generation of the facility. For example, for a fuel powered electrical generation facility, the market price path for fuel over the period of time can influence optimal scheduling. In addition, the market price path, over the period of time, for electricity, as generated by the facility, can also influence optimal scheduling.
Another set of factors can include constraints associated with operation of the facility. For example, electrical power generation facilities are generally subject to physical constraints including maximum and minimum generation rates while the facility is operating, maximum and minimum operating times, or run-times, and maximum generation rate increases and decreases per unit of time (herein termed, respectively, maximum ramp up and ramp down rates). In addition, other constraints can relate to costs associated with specific scheduling actions or conduct under specific circumstances, such as start-up costs and shutdown costs. For example, start-up costs can include the cost of a necessary quantity of a certain type of fuel used for initiation of operation, which can be different and more expensive than a primary operation fuel. The effect of these various constraints on formulating optimal scheduling is sometimes referred to as the “unit commitment” problem.
Further, in some situations, it is attempted to optimize the schedules of, and value, each of a group of facilities, with constraints that apply to the group as a whole, such as, for example, a cumulative generation quota applying to the sum total generation output of the group. In addition, in some group optimization and valuation situations, price paths for commodities, such as fuel and electricity, can be influenced by the operation of each facility in the group, which is sometimes referred to as a liquidity problem. In such cases, optimal scheduling of each facility can depend in part on operation of each of the other facilities in the group.
As demonstrated by the above, optimal scheduling of one or more generation facilities over a period of time requires informed decision-making over the period of time, taking into account any relevant prices at particular times throughout the period, e.g. price paths, as well as constraints as they apply at particular times throughout the period, including operating limits and costs associated with certain operating actions. Further complicating matters, some constraints can themselves be dependent on previous operation scheduling. It must also be noted that optimal scheduling at a particular time during the period can, and must, take into account circumstances as they have developed prior to, and exist at, that particular time, since an actual operator would be aware of such circumstances. Optimal scheduling at a given time further requires taking into account the current forecasts and probabilities as to relevant future circumstances, including relevant price paths, of which an operator would also be aware. Herein, “optimal scheduling” includes scheduling which is optimized in the sense of an actual operator making optimal decisions over the period of time.
Given the complexity of determining optimal scheduling, determining an anticipated profitability is naturally also complex. To assess the profitability, and hence the value, of a generation facility over a future period of time, optimal scheduling decision-making is to be assumed over that period of time. Optimal scheduling over a period of time, however, can only be realistically assessed by considering, at various different times throughout the period of time, the information available to, and the circumstances confronting, an operator at each of the various different times, such as, for example, at each hour interval or day interval. To assume that an operator has exact knowledge of future circumstances is, of course, unrealistic, just as it is unrealistic to assume that an operator is unaware of past and present circumstances. As such, to be accurate, assessments of optimal scheduling over a period of time, from the point of view of an operator during the period of time, must take into account, at various particular times over the period, optimal scheduling decision-making at each of various times, assuming knowledge of present circumstances by the operator and projections regarding future circumstances, as such circumstances and decision-making evolves throughout the period.
It is well recognized in the art that valuation and optimization of scheduling of facilities such as electrical power generation facilities present important and complex challenges, and numerous approaches and techniques have been developed to attempt to address such problems. Some such approaches apply financial instrument forms or financial modeling techniques to the power generation context. For example, some approaches, such as spark spread option modeling, use stochastic, or probabilistic, techniques to model relevant price paths. Some varieties, such as enhanced or “swing” spark spread option modeling, used in European style securities, can include the use of Monte Carlo simulations to randomly generate data such as relevant price paths in accordance with a specified forecast and probabilistic characteristics.
While spark spread option modeling takes into account forecasts and specified uncertainty with regard to such factors as relevant price paths, facility constraints, examples of which are discussed above, are not modelable using such techniques. In decision making with regard to trading in financial instruments whose value depends on the value of a generation facility, spark spread option modeling is sometimes used in combination with “hedging” strategies. Hedging strategies, which can involve trading in certain financial instruments such as futures contracts, are used to attempt to reduce risk when investing in markets involving a specified estimated or forecasted uncertainty. Hedging strategies, however, do not remedy or even address the accuracy, and therefore the degree of value, of the underlying technique by which values of instruments are calculated. Spark spread option modeling, whether or not used with hedging, does not take into account relevant constraints, and, as such, cannot provide accurate valuation models for generation facilities and other facilities, nor can such modeling techniques be used to optimize scheduling.
Another category of approaches used in the context of generation facility valuation and scheduling optimization is traditional dynamic programming. In this context, traditional dynamic programming techniques typically use a backward iteration algorithm, sometimes through a mathematical decision tree, to attempt to optimize generation facility scheduling over a period of time divided into a number of time intervals. In combination with input constraints and forecasted price paths, traditional dynamic programming can be used to attempt to value generation facilities and financial instruments associated therewith. Constraints are typically modeled by including a variable in the dynamic programming optimization formulas to represent the constraint. By dividing a time period into a discrete number of evaluated intervals rather than evaluating the time period as a continuous period, the potentially formidable mathematics involved in traditional dynamic programming can be made somewhat less intractable. Even with such interval division, however, as the number of explicitly modeled constraints grows, the mathematics become increasingly complex at a rate geometrically proportional to the number of explicitly modeled variables. This complexity can be explained by the fact that each explicitly modeled variable increases the dimensionality of a theoretical mathematical “surface” on which is plotted the range of possible scenarios contemplated by the traditional dynamic programming method. For this reason, the computational unmanageability of explicitly modeling many constraints is known as the “curse of dimensionality.”
Some traditional dynamic programming techniques use, as input, mathematically formulated constraints and a specified price path for fuel and electricity, and use backward iteration to attempt to find a profit-maximizing operation schedule. Traditional dynamic programming, however, is deterministic in that it fails to take into account uncertainty with regard to relevant future price paths. Traditional dynamic programming effectively attempts to determine an optimal schedule for a facility assuming perfect and sure knowledge of future prices, which is, of course, unrealistic.
In addition, existing techniques may not take into account certain types of heat rate functions, or heat rate curves. A heat rate curve can specify the efficiency of a generation facility as a function of the facility's generation rate. Existing techniques may only be capable of handling convex or monotonically increasing heat rate curves. In reality, however, generators are often characterized by decreasing heat rate curves.
In recognition of the incomplete adequacy of spark spread option modeling and of traditional dynamic programming by themselves, methods have been developed which attempt to combine the stochastic price path modeling advantages of spark spread option modeling with the backward iterative scheduling optimization approach of dynamic programming. One such method proceeds as follows. First, based on specified forecasted price paths, traditional dynamic programming is used to generate what is considered to be an optimal operation schedule over a period. Next, given a specified uncertainty about the forecasted price path, a set of possible price paths are generated, by Monte Carlo simulation or otherwise. Next, the previously determined optimal schedule is applied to each of the price path possibilities. Profit or loss for the facility is calculated for each price path scenario, and may be calculated by summing profit or loss at each of the time intervals. Averaging profit or loss over all of the scenarios yields an anticipated profitability over the period.
One problem with the foregoing method, however, is that the same originally determined schedule is applied in each price path scenario. In reality, the determined schedule has only been optimized for the single original forecasted price path scenario. As such, the schedule is not optimized for the various generated price path scenarios. In reality, for each different price path scenario, an operator would have the opportunity to adjust scheduling throughout the period, based on evolving price paths or other variables, to attempt to maximize profit in that particular scenario. By effectively assuming an operator who is oblivious to changing price paths and other variables, this method undervalues the profitability of the facility. Ultimately, therefore, using a set schedule over various price path scenarios is unrealistic and inaccurate.
Another method which attempts to combine the advantages of spark spread option based modeling and dynamic programming proceeds as follows. First, based on a forecasted price path and an anticipated specified uncertainty with regard to the price path, a set of possible price paths are generated, using Monte Carlo simulations or otherwise. Next, for each price path scenario, traditional dynamic programming is used to determine what is considered to be an optimal schedule for each price path scenario. Next, profit or loss is determined for each price path scenario using, for each price path scenario, the optimal schedule for that particular scenario.
One problem with the foregoing method is that, for each different price path scenario, the traditional dynamic programming technique used models a situation in which the exact price paths for the entire period are known throughout the entire period. As such, the foregoing method effectively assumes that the operator has complete and exact knowledge of future price paths. By using unrealistically optimized schedules, the foregoing method overvalues the profitability of the facility over the period of time.
In summary, despite the above described attempts at combining the advantages of both traditional dynamic programming and spark spread option modeling, none of the existing methods for valuation and optimization of scheduling of generation facilities succeeds in providing accurate results. In reality, an operator has knowledge of past and present circumstances, including past and present scheduling as well as past and present price paths. With regard to anticipated future price paths, the best an operator can do is to use presently available information for stochastic projections, or forecasts, of price paths or other variables, which can include, for example, a forecasted price path and a specified projected level of uncertainty over time with regard thereto. Such stochastic projections will of course change at each successive time of evaluation, as price paths and scheduling paths continually evolve. Consequently, evaluation regarding an optimal scheduling action at each successive interval must be determined based on the exact price path and scheduling path knowledge as it has evolved up to that time, combined with the stochastic projection information available at that time.
References in this and related technical areas are representative of the state of the art as described above. U.S. Pat. No. 5,974,403, issued on Oct. 26, 1999 to Takriti et al., discusses a computerized method for forecasting electrical power spot price and for forecasts related to power trading. Employing stochastic price path forecasting techniques and discussing hedging techniques, U.S. Pat. No. 5,974,403 does not adequately model various dynamic scheduling factors.
U.S. Pat. No. 6,021,402, issued on Feb. 1, 2000 to Takriti, discusses scheduling for generators for risk management purposes. Multiple price path scenarios are generated, and decision-tree based dynamic programming is applied to optimize scheduling in each scenario.
The article, “Interruptible Physical Transmission Contracts for Congestion Management,” Santosh Raikar and Marija Ilic, Energy Laboratory Publication No. MIT EL 01-010WP, Energy Laboratory, Massachusetts Institute of Technology, February 2001, discusses trinomial tree based traditional dynamic programming techniques as applied to transmission contracts, as well as enhanced spark spread option stochastic techniques, but does not adequately model various constraints in a dynamic, stochastic context.
The article, “Valuation of Generation Assets with Unit Commitment Constraints under Uncertain Fuel Prices,” Petter Skantze, Poonsaeng Visudhiphan, and Marija Ilic, Energy Laboratory Publication No. MIT EL 00-006, Energy Laboratory, Massachusetts Institute of Technology, November 2000, discusses the use of multinomial trees and stochastic price path forecasting techniques, generating a set of price path scenarios and applying dynamic programming to each.
The Article, “The Challenges of Pricing and Risk Management of Energy Derivatives,” Les Clewlow, AC3 Financial Engineering Using High Performance Computing, Nov. 27, 2001, discusses stochastic price path forecasting techniques, but does not adequately model various constraints in a dynamic, stochastic context.
The article, “Financial Risk Management in a Competitive Electricity Market,” Roger Bjorgan, Chen-Ching Lui, Jacqes Lawarree, University of Washington, discusses dynamic programming and the use of hedging techniques.
The Master's Thesis, Decision Tools for Electricity Transmission Service and Pricing: a Dynamic Programming Approach,” Ozge Nadia Gozum, Massachusetts Institute of Technology, 2000, discusses use of Monte Carlo simulation to generate a set of price paths, and application of dynamic programming to each.
In summary, references in the field demonstrate a recognition of the need to combine the advantages of dynamic programming with the advantages of stochastic price path modeling to realistically value and optimize scheduling of a facility, such as a generation facility. The state of the art, however, has not succeeded in providing a method to effectively combine the two. In addition, the state of the art does not provide a method to overcome the curse of dimensionality inherent in explicit modeling of numerous constraints in traditional dynamic programming.
For all of the above reasons, there is a need in the art for a method for valuing and optimizing scheduling of facilities, such as generation facilities, that effectively combines the advantages of traditional dynamic programming with the advantages of stochastic price path modeling. In addition, there is a need in the art for a method which realizes the advantages of traditional dynamic programming while avoiding mathematical intractability caused by the “curse of dimensionality.”